If G is a looped graph, then its adjacency matrix represents a binary
matroid MAβ(G) on V(G). MAβ(G) may be obtained from the delta-matroid
represented by the adjacency matrix of G, but MAβ(G) is less sensitive to
the structure of G. Jaeger proved that every binary matroid is MAβ(G) for
some G [Ann. Discrete Math. 17 (1983), 371-376].
The relationship between the matroidal structure of MAβ(G) and the
graphical structure of G has many interesting features. For instance, the
matroid minors MAβ(G)βv and MAβ(G)/v are both of the form
MAβ(Gβ²βv) where Gβ² may be obtained from G using local
complementation. In addition, matroidal considerations lead to a principal
vertex tripartition, distinct from the principal edge tripartition of
Rosenstiehl and Read [Ann. Discrete Math. 3 (1978), 195-226]. Several of these
results are given two very different proofs, the first involving linear algebra
and the second involving set systems or delta-matroids. Also, the Tutte
polynomials of the adjacency matroids of G and its full subgraphs are closely
connected to the interlace polynomial of Arratia, Bollob\'{a}s and Sorkin
[Combinatorica 24 (2004), 567-584].Comment: v1: 19 pages, 1 figure. v2: 20 pages, 1 figure. v3:29 pages, no
figures. v3 includes an account of the relationship between the adjacency
matroid of a graph and the delta-matroid of a graph. v4: 30 pages, 1 figure.
v5: 31 pages, 1 figure. v6: 38 pages, 3 figures. v6 includes a discussion of
the duality between graphic matroids and adjacency matroids of looped circle
graph